Cartographic Projection
every map is wrong, and by exactly how much
Field

What each projection optimises

Every projection minimises something. Naming the objective is more honest than naming the shape of the paper it was notionally rolled from.
0°20°40°60°80°MercatorGall–PetersWinkelRobinsonlatitudeangular deformationalong a meridian

Every projection minimises something

A projection is the solution to an optimisation problem, and naming the objective explains more than naming the family. Some objectives are exact constraints, some are least-squares fits, and one is a table of numbers a man adjusted until it looked right.

 EquirectangularMercatorMollweideSinusoidalRobinsonWinkel tripelsame sphere, same graticuleno two agree

Which projection is best

An incomplete question, and the incompleteness is the answer. Every projection preserves something and destroys something else, so the comparison worth making is between a projection and a purpose, not between two projections.

 RobinsonWinkel tripelMollweideMercatorGall–PetersEckert IVsame sphere, same graticuleno two agree

Compromise projections

A projection that preserves nothing exactly can distort everything less than one that preserves something exactly. For a general-purpose world map that is the right trade, and it is why the two most widely used ones today have no exact property at all.

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